{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:OVPESULMWGEJYEKESSEBJRUS3Q","short_pith_number":"pith:OVPESULM","canonical_record":{"source":{"id":"1301.6235","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-26T11:09:40Z","cross_cats_sorted":[],"title_canon_sha256":"22e0ddf7fcb5598283cd60448f6ddef63045155b8e1a38e4be39ea58df3b2d51","abstract_canon_sha256":"0d504ce46380f12955327ed1469be530c849d1d69760bb08f70920267d8e2f88"},"schema_version":"1.0"},"canonical_sha256":"755e49516cb1889c1144948814c692dc3154323cc86271d48162983444a564c6","source":{"kind":"arxiv","id":"1301.6235","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.6235","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"arxiv_version","alias_value":"1301.6235v2","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6235","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"pith_short_12","alias_value":"OVPESULMWGEJ","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OVPESULMWGEJYEKE","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OVPESULM","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:OVPESULMWGEJYEKESSEBJRUS3Q","target":"record","payload":{"canonical_record":{"source":{"id":"1301.6235","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-26T11:09:40Z","cross_cats_sorted":[],"title_canon_sha256":"22e0ddf7fcb5598283cd60448f6ddef63045155b8e1a38e4be39ea58df3b2d51","abstract_canon_sha256":"0d504ce46380f12955327ed1469be530c849d1d69760bb08f70920267d8e2f88"},"schema_version":"1.0"},"canonical_sha256":"755e49516cb1889c1144948814c692dc3154323cc86271d48162983444a564c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:37.598931Z","signature_b64":"ZeWZOLLM/OACZbluG1mp4ohyy4+rt2A3YGOmzL1xirbHcyhm44CrgFbR2MVBEc/n8H+NgaTrv5uZY4Zhz4IWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"755e49516cb1889c1144948814c692dc3154323cc86271d48162983444a564c6","last_reissued_at":"2026-05-18T03:34:37.598367Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:37.598367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.6235","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:34:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1SEWBkNMJ6XVk0oDKifFwcG+ELVWlXizkBIomwB+65xNd23TahGFN1wDjZvDw8PPXuadwMHbRYIpDPe9T/q4BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:10:59.222382Z"},"content_sha256":"450d51a5d0eddff755b6b9ce974e3df5a16d6a879fe3fecfac05c8d82745edc4","schema_version":"1.0","event_id":"sha256:450d51a5d0eddff755b6b9ce974e3df5a16d6a879fe3fecfac05c8d82745edc4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:OVPESULMWGEJYEKESSEBJRUS3Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp criteria of Liouville type for some nonlinear systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Congming Li, Yutian Lei","submitted_at":"2013-01-26T11:09:40Z","abstract_excerpt":"In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6235","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:34:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"587zzxTLgbnCbUMZMN1Zp9pUOasoFqyPzcBEoqDa2ggn5avsL3WDaC4Ic5QdfBWxmNVZI3UWzEh6ijL7nj5MBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T00:10:59.246629Z"},"content_sha256":"85508529b1a8083d9f5b11869547950a4ec0ec66cb0829c2128aaef25be092c4","schema_version":"1.0","event_id":"sha256:85508529b1a8083d9f5b11869547950a4ec0ec66cb0829c2128aaef25be092c4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OVPESULMWGEJYEKESSEBJRUS3Q/bundle.json","state_url":"https://pith.science/pith/OVPESULMWGEJYEKESSEBJRUS3Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OVPESULMWGEJYEKESSEBJRUS3Q/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-23T00:10:59Z","links":{"resolver":"https://pith.science/pith/OVPESULMWGEJYEKESSEBJRUS3Q","bundle":"https://pith.science/pith/OVPESULMWGEJYEKESSEBJRUS3Q/bundle.json","state":"https://pith.science/pith/OVPESULMWGEJYEKESSEBJRUS3Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OVPESULMWGEJYEKESSEBJRUS3Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:OVPESULMWGEJYEKESSEBJRUS3Q","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0d504ce46380f12955327ed1469be530c849d1d69760bb08f70920267d8e2f88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-26T11:09:40Z","title_canon_sha256":"22e0ddf7fcb5598283cd60448f6ddef63045155b8e1a38e4be39ea58df3b2d51"},"schema_version":"1.0","source":{"id":"1301.6235","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.6235","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"arxiv_version","alias_value":"1301.6235v2","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6235","created_at":"2026-05-18T03:34:37Z"},{"alias_kind":"pith_short_12","alias_value":"OVPESULMWGEJ","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OVPESULMWGEJYEKE","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OVPESULM","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:85508529b1a8083d9f5b11869547950a4ec0ec66cb0829c2128aaef25be092c4","target":"graph","created_at":"2026-05-18T03:34:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we es","authors_text":"Congming Li, Yutian Lei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-26T11:09:40Z","title":"Sharp criteria of Liouville type for some nonlinear systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6235","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:450d51a5d0eddff755b6b9ce974e3df5a16d6a879fe3fecfac05c8d82745edc4","target":"record","created_at":"2026-05-18T03:34:37Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0d504ce46380f12955327ed1469be530c849d1d69760bb08f70920267d8e2f88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-01-26T11:09:40Z","title_canon_sha256":"22e0ddf7fcb5598283cd60448f6ddef63045155b8e1a38e4be39ea58df3b2d51"},"schema_version":"1.0","source":{"id":"1301.6235","kind":"arxiv","version":2}},"canonical_sha256":"755e49516cb1889c1144948814c692dc3154323cc86271d48162983444a564c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"755e49516cb1889c1144948814c692dc3154323cc86271d48162983444a564c6","first_computed_at":"2026-05-18T03:34:37.598367Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:37.598367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZeWZOLLM/OACZbluG1mp4ohyy4+rt2A3YGOmzL1xirbHcyhm44CrgFbR2MVBEc/n8H+NgaTrv5uZY4Zhz4IWCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:37.598931Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.6235","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:450d51a5d0eddff755b6b9ce974e3df5a16d6a879fe3fecfac05c8d82745edc4","sha256:85508529b1a8083d9f5b11869547950a4ec0ec66cb0829c2128aaef25be092c4"],"state_sha256":"bf0359fb1b4262973aa362924242add4b305eafe6a986a985b77141538a1d686"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"riTwaBCoVjaEVsSJUSIuJz7BF48nBmesoSY6O3PmNmRjihX6dauVpcWWcCg5pncL62l3lHCFt4856S8SvdXQDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T00:10:59.252637Z","bundle_sha256":"35276caac54330137ff36117159d0ea5adeb1001042b889827379d099cb043cc"}}